If $\lbrace X_{n}\rbrace$ is a submartingale with respect to filtration $\lbrace \mathcal{F}_{n}\rbrace$, and $\phi:\mathbb{R}\to\mathbb{R}$ is convex and increasing, then $\lbrace\phi(X_{n})\rbrace$ is a submartingale, too.
- By convexity of $\phi$ and the fact that $X_{n}$ is $\mathcal{F}_{n}$-measurable, $\phi(X_{n})$ is $\mathcal{F}_{n}$-measurable, thus the new process is adapted to $\lbrace \mathcal{F}_{n}\rbrace$.
- To verify that $\mathbb{E}(\phi(X_{n+1})|\mathcal{F}_{n})\geq\phi(X_{n})$, Jensen's inequality can be applied.
- Now my question: How to show that $\mathbb{E}(\phi(X_{n})^{+})<\infty$?
